The windoze app that comes most quickly (no pun) to mind is Quicken - a friend of mine went to a WINE site that lists each app and how it works under WINE and it got an "OK but still buggy and some features don't always work right" rating - by that I assume that you could start it up and do some basic stuff but it wouldn't be an equivalent experience to running it on windoze - I haven't tried it myself however. I'm sure things will only improve and I can't imagine that Quicken does anything too exotic under windoze - it's OS calls I would think would be pretty basic - although I guess it probably has some hooks in to IE, which might be an issue.
Depending on how old your version of Quicken is, you'd probably want to try it out under the latest version of Wine. It seems to be fairly decent from what I can tell.
Also, if you're familiar with financial software, check out GnuCash--it looks like it's Quicken-compatible.